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Zorluk: OrtaMatrices and Determinants

If A=(2314)A = \begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix} and B=(510−2)B = \begin{pmatrix} 5 & 1 \\ 0 & -2 \end{pmatrix}, what is the product matrix ABAB?

  1. (10−45−7)\begin{pmatrix} 10 & -4 \\ 5 & -7 \end{pmatrix}Cevap
  2. B
    (1030−8)\begin{pmatrix} 10 & 3 \\ 0 & -8 \end{pmatrix}
  3. C
    (1119−2−8)\begin{pmatrix} 11 & 19 \\ -2 & -8 \end{pmatrix}
  4. D
    (1085−7)\begin{pmatrix} 10 & 8 \\ 5 & -7 \end{pmatrix}

Cevap

The matrix product ABAB is (10−45−7)\begin{pmatrix} 10 & -4 \\ 5 & -7 \end{pmatrix}.
Matrix multiplication is evaluated using the row-by-column rule. Evaluating each entry: top-left is 2(5)+3(0)=102(5)+3(0)=10, top-right is 2(1)+3(−2)=−42(1)+3(-2)=-4, bottom-left is 1(5)+4(0)=51(5)+4(0)=5, and bottom-right is 1(1)+4(−2)=−71(1)+4(-2)=-7. Thus, the product matrix is (10−45−7)\begin{pmatrix} 10 & -4 \\ 5 & -7 \end{pmatrix}.

Adım Adım Çözüm

1
Calculate the top-left element (Row 1 of A ⋅\cdot Column 1 of B)
2(5)+3(0)=10+0=102(5) + 3(0) = 10 + 0 = 10
Matrix multiplication takes the dot product of the corresponding row of the first matrix and column of the second matrix.
2
Calculate the top-right element (Row 1 of A ⋅\cdot Column 2 of B)
2(1)+3(−2)=2−6=−42(1) + 3(-2) = 2 - 6 = -4
Combine products of elements in row 1 of A and column 2 of B.
3
Calculate the bottom-left element (Row 2 of A ⋅\cdot Column 1 of B)
1(5)+4(0)=5+0=51(5) + 4(0) = 5 + 0 = 5
Combine products of elements in row 2 of A and column 1 of B.
4
Calculate the bottom-right element (Row 2 of A ⋅\cdot Column 2 of B)
1(1)+4(−2)=1−8=−71(1) + 4(-2) = 1 - 8 = -7
Combine products of elements in row 2 of A and column 2 of B.

Anahtar Kavram

Matrix Multiplication (Row-by-Column Rule)
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Konular

Matrices and Determinants

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